```ts
/**
 * a divided by b, throwing an Error if b is zero or if either a or b is not a finite number.
 */
export function safeDivide(a: number, b: number): number {
  if (typeof a !== 'number' || typeof b !== 'number' || !Number.isFinite(a) || !Number.isFinite(b)) {
    throw new Error('Both arguments must be finite numbers.');
  }
  if (b === 0 || b === -0) {
    throw new Error("Cannot divide by zero.");
  }
  return a / b;
}
```